Onp-absolutely summing constants of banach spaces

Given 1≦p<∞ and a real Banach spaceX, we define thep-absolutely summing constantμ _p(X) as inf{Σ _i ^=1/m |x*(x _i)|^p p Σ _i ^=1/m ‖x _i‖^p p]¹ p}, where the supremum ranges over {x*∈X*; ‖x*‖≤1} and the infimum is taken over all sets {x ₁,x ₂, …,x _m} ⊂X such that Σ _i ^=1/m ‖x _i‖>0. It follows immediately from [2] thatμ _p(X)>0 if and only ifX is finite dimensional. In this paper we find the exact values ofμ _p(X) for various spaces, and obtain some asymptotic estimates ofμ _p(X) for general finite dimensional Banach spaces. This is a part of the author’s Ph.D. Thesis prepared at the Hebrew University of Jerusalem, under the supervision of Prof. A. Dvoretzky and Prof. J. Lindenstrauss.     

On nonrational divisors over non-Gorenstein terminal singularities

Let (X, o) be a germ of a 3-dimensional terminal singularity of index m ≥ 2. If (X, o) has type cAx/4, cD/3-3, cD/2-2, or cE/2, then we assume that the standard equation of X in ℂ⁴/ℤ _m is nondegenerate with respect to its Newton diagram. Let π: Y → X be a resolution. We show that there are at most 2 nonrational divisors E _i , i = 1, 2, on Y such that π(E _i ) = o and the discrepancy a(E _i , X) is at most 1. When such divisors exist, we describe them as exceptional divisors of certain blowups of (X, o) and study their birational type. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 2, pp. 169–184, 2005.     

Random matrix products and applications to cellular automata

Let λ be the upper Lyapunov exponent corresponding to a product of i.i.d. randomm×m matrices (X _i) _i ^0/∞ over ℂ. Assume that theX _i's are chosen from a finite set {D ₀,D ₁...,D _t-1(ℂ), withP(X _i=D_j)>0, and that the monoid generated byD ₀, D₁,…, D_q−1 contains a matrix of rank 1. We obtain an explicit formula for λ as a sum of a convergent series. We also consider the case where theX _i's are chosen according to a Markov process and thus generalize a result of Lima and Rahibe [22]. Our results on λ enable us to provide an approximation for the numberN _≠0(F(x)ⁿ,r) of nonzero coefficients inF(x) ⁿ.(modr), whereF(x) ∈ ℤ[x] andr≥2. We prove the existence of and supply a formula for a constant α (<1) such thatN _≠0(F(x)ⁿ,r) ≈n ^α for “almost” everyn. Supported in part by FWF Project P16004-N05     

On Projections of Semi-Algebraic Sets Defined by Few Quadratic Inequalities

Let S⊂ℝ ^k+m be a compact semi-algebraic set defined by P ₁≥0,…,P _ℓ ≥0, where P _i ∈ℝ[X ₁,…,X _k ,Y ₁,…,Y _m ], and deg (P _i )≤2, 1≤i≤ℓ. Let π denote the standard projection from ℝ ^k+m onto ℝ ^m . We prove that for any q>0, the sum of the first q Betti numbers of π(S) is bounded by (k+m) ^{O(q ℓ)}. We also present an algorithm for computing the first q Betti numbers of π(S), whose complexity is . For fixed q and ℓ, both the bounds are polynomial in k+m. The author was supported in part by an NSF Career Award 0133597 and a Sloan Foundation Fellowship.     

Regression of polynomial statistics on the sample mean and natural exponential families

A polynomial Q = Q(X ₁, …, X _n ) of degree m in independent identically distributed random variables with distribution function F is an unbiased estimator of a functional q(α ₁(F), …, α _m (F)), where q(u ₁, …, u _m ) is a polynomial in u ₁, …, u _m and α _j (F) is the jth moment of F (assuming the necessary moment of F exists). It is shown that the relation E(Q | X ₁ + … + X n) = 0 holds if and only if q(α ₁(θ), …, α _m (θ)) ≡ 0, where α _j (θ) is the jth moment of the natural exponential family generated by F. This result, based on the fact that X ₁ + … + X n is a complete sufficient statistic for a parameter θ in a sample from a natural exponential family of distributions F _θ(x) = ∫_−∞ ^x e ^θu−k(θ) dF(u), explains why the distributions appearing as solutions of regression problems are the same as solutions of problems for natural exponential families though, at the first glance, the latter seem unrelated to the former.     

Functional characterization of Vasil’ev invariants

A cover of a manifold X is called an r-cover if any r points of X belong to a set in the cover. Let X and Y be two smooth manifolds, let Emb(X, Y) be the family of smooth embeddings X → Y, let M be an Abelian group, and let F: Emb(X, Y) → M be a functional. One says that the degree of F does not exceed r if for each finite open r-cover {U _i } _i∈I ; of X there exist functionals F _i : Emb(U _i , Y) → M, i ∈ I, such that for each a ∈ Emb(X, Y) one has

Higher Chern Classes and Steenrod Operations in Motivic Cohomology

In this short paper we investigate the relation between higher Chern classes and reduced power operations in motivic cohomology. More precisely, we translate the well-known arguments [5] into the context of motivic cohomology and define higher Chern classes c_p,q : K _p(X) → H^2q-p (X,Z(q)) → H^2q-p(X, Z/l(q)), where X is a smooth scheme over the base field k, l is a prime number and char(k) ≠ l. The same approach produces the classes for K-theory with coefficients as well. Let further Pⁱ : H^m(X, Z/l(n)) → H^m+2i(l-1) (X, Z/l(n + i(l - 1))) denote the ith reduced power operation in motivic cohomology, constructed in [2]. The main result of the paper looks as follows.     

Elementary Symmetric Polynomials in Random Variables

The subject of the paper is the probability-theoretic properties of elementary symmetric polynomials σ _k of arbitrary degree k in random variables X _i (i=1,2,…,m) defined on special subsets of commutative rings ℛ _m with identity of finite characteristic m. It is shown that the probability distributions of the random elements σ _k (X ₁,…,X _m ) tend to a limit when m→∞ if X ₁,…,X _m form a Markov chain of finite degree μ over a finite set of states V, V⊂ℛ _m , with positive conditional probabilities. Moreover, if all the conditional probabilities exceed a prescribed positive number α, the limit distributions do not depend on the choice of the chain.      

Chaos caused by a strong-mixing measure-preserving transformation

The chaos caused by a strong-mixing preserving transformation is discussed and it is shown that for a topological spaceX satisfying the second axiom of countability and for an outer measurem onX satisfying the conditions: (i) every non-empty open set ofX ism-measurable with positivem-measure; (ii) the restriction ofm on Borel σ-algebra ℬ(X) ofX is a probability measure, and (iii) for everyY⊂X there exists a Borel setB⊂ℬ(X) such thatB⊃Y andm(B) =m(Y), iff:X→X is a strong-mixing measure-preserving transformation of the probability space (X, ℬ(X),m), and if {m}, is a strictly increasing sequence of positive integers, then there exists a subsetC⊂X withm (C) = 1, finitely chaotic with respect to the sequence {m _i}, i.e. for any finite subsetA ofC and for any mapF:A→X there is a subsequencer _i such that lim_i→∞ f ^r i(a) =F(a) for anya ∈A. There are some applications to maps of one dimension. the National Natural Science Foundation of China.     

On moduli of vector bundles on surfaces with negative Kodaira dimension

Summary Here we prove the following result. Fix integersq, τ,a’, b’, a’ _i, 1≤i≤τ,a’, b’, a’ _i, 1≤i≤τ; then there is an integerew such that for every integert≥w, for every algebraically closed fieldK for every smooth complete surfaceX with negative Kodaira dimension, irregularityq andK _X ² =8(1−q)−τ, the following condition holds; ifX→S is a sequence fo τ blowing-downs which gives a relatively minimal model with ruling ρ:S→C, take as basis of the Neron Severi groupNS(X) a smooth rational curve which is the total transform of a fiber ofC, the total transform of a minimal section of ρ and the total transformD _i, 1≤i≤τ, of the exceptional curver; then for everyH andL∈Pic (X) withH ample,H (resp.L) represented by the integersa’, b’, a’ _i, (resp.a’, b’, a’ _i), 1≤i≤τ, in the chosen basis ofNS(X) the moduli spaceM(ZX, 2,H, L, t) of rank 2H-stable vector bundles onX with determinantL andc ₂=t is generically smooth and the number, dimension and ?birational structure? of the irreducible components ofM(X, 2,H, L, t)_red do not depend on the choice ofK andX. Furthermore the birational structure of these irreducible components can be loosely described in terms of the birational structure of the components of suitableM(S, 2,H’, L’, t’)_red withS a relatively minimal model ofX.

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Sunto SiaX una superficie algebrica liscia completa con dimensione di Kodaira negativa e definita su un campo algebricamente chiusoK; fissiamoH eL∈Pic (X),t∈Z; siaq l’irregolarità diX e τ≔8(1−q)−K _X ^Emphasis>2 ; siaM(X, 2,H, L, t) to schema dei moduli dei fibrati vettorialiH-stabili di rango 2 suX con determinateL ec ₂=t. Si dimostra che esiste una costantew che dipende solo daq, da τ e dalla classe numerica diH e diL (ma non da char (K) o dalla classe di isomorphismo diX) tale che per ognit≥w il numero, la dimensione e ?la struttura birazionale? delle componenti irriducibili diM(X, 2,H, L, t)_red non dipende dalla scelta di char (K),K eX ma solo daq, τ e dalle classi diH eL inNS(X). Inoltre la ?struttura birazionale? di queste componenti irriducibili può essere grossolanamente descritta in termini delle componenti di opportuni spazi di moduliM(S, 2,H’, L’, t’) (doveS è un modello minimale diX).

     

On some new generalized difference sequence spaces defined by a sequence of moduli

In this paper, we define the new generalized difference sequence spaces [V, λ, F, p, q]₀(Δ _v ^m ), [V, λ, F, p, q]₁(Δ _v ^m ) and [V, λ, F, p, q]_∞(Δ _v ^m ). We also study some inclusion relations between these spaces.     

Group-theoretical generalization of necklace polynomials

Let G be a group, U a subgroup of G of finite index, X a finite alphabet and q an indeterminate. In this paper, we study symmetric polynomials M _G (X,U) and M_G^q(X,U)M_{G}^{q}(X,U) which were introduced as a group-theoretical generalization of necklace polynomials. Main results are to generalize identities satisfied by necklace polynomials due to Metropolis and Rota in a bijective way, and to express M_G^q(X,U)M_{G}^{q}(X,U) in terms of M _G (X,V)’s, where [V] ranges over a set of conjugacy classes of subgroups to which U is subconjugate. As a byproduct, we provide the explicit form of the GL _m (ℂ)-module whose character is M_\mathbbZ^q(X,n\mathbbZ)M_{\mathbb{Z}}^{q}(X,n\mathbb{Z}), where m is the cardinality of X.     

Graph-Theoretic Approach to Stochastic Integrals with Clifford Algebras

Given a fixed probability space (Ω,ℱ,ℙ) and m≥1, let X(t) be an L²(Ω) process satisfying necessary regularity conditions for existence of the mth iterated stochastic integral. For real-valued processes, these existence conditions are known from the work of D. Engel. Engel’s work is extended here to L²(Ω) processes defined on Clifford algebras of arbitrary signature (p,q), which reduce to the real case when p=q=0. These include as special cases processes on the complex numbers, quaternion algebra, finite fermion algebras, fermion Fock spaces, space-time algebra, the algebra of physical space, and the hypercube. Next, a graph-theoretic approach to stochastic integrals is developed in which the mth iterated stochastic integral corresponds to the limit in mean of a collection of weighted closed m-step walks on a growing sequence of graphs. Combinatorial properties of the Clifford geometric product are then used to create adjacency matrices for these graphs in which the appropriate weighted walks are recovered naturally from traces of matrix powers. Given real-valued L²(Ω) processes, Hermite and Poisson-Charlier polynomials are recovered in this manner.     

Exact laws for sums of ratios of order statistics from the Pareto distribution

Consider independent and identically distributed random variables {X _nk, 1 ≤ k ≤ m, n ≤ 1} from the Pareto distribution. We select two order statistics from each row, X _n(i) ≤ X _n(j), for 1 ≤ i < j ≤ = m. Then we test to see whether or not Laws of Large Numbers with nonzero limits exist for weighted sums of the random variables R _ij = X _n(j)/X _n(i).     

Minimum and Surjectivity Moduli Preserving in Banach Space

Let m(T) and q(T) be respectively the minimum and the surjectivity moduli of T∈ℬ(X), where ℬ(X) denotes the algebra of all bounded linear operators on a complex Banach space X. If there exists a semi-invertible but non-invertible operator in ℬ(X) then, given a surjective unital linear map φ: ℬ(X)⟶ℬ(X), we prove that m(T)=m(φ(T)) for all T∈ℬ(X), if and only if, q(T)=q(φ(T)) for all T∈ℬ(X), if and only if, there exists a bijective isometry U∈ℬ(X) such that φ(T)=UTU ⁻¹ for all T∈ℬ(X).     

Feynman’s Operational Calculi: Decomposing Disentanglings

Let X be a Banach space and suppose that A ₁,…,A _n are noncommuting (that is, not necessarily commuting) elements in ℒ(X), the space of bounded linear operators on X. Further, for each i∈{1,…,n}, let μ _i be a continuous probability measure on ℬ([0,1]), the Borel class of [0,1]. Each such n-tuple of operator-measure pairs (A _i ,μ _i ), i=1,…,n, determines an operational calculus or disentangling map T_m1,...,mn{\mathcal{T}}_{\mu_{1},\dots,\mu_{n}} from a commutative Banach algebra \mathbbD(A₁,...,A_n){\mathbb{D}}(A_{1},\dots,A_{n}) of analytic functions, called the disentangling algebra , into the noncommutative Banach algebra ℒ(X). The disentanglings are the central processes of Feynman’s operational calculi.     

On the multiplicity of the eigenvalues of a graph

Given a graph G with characteristic polynomial ϕ(t), we consider the ML-decomposition ϕ(t) = q ₁(t)q ₂(t)² ... q _m (t)^m, where each q _i (t) is an integral polynomial and the roots of ϕ(t) with multiplicity j are exactly the roots of q _j (t). We give an algorithm to construct the polynomials q _i (t) and describe some relations of their coefficients with other combinatorial invariants of G. In particular, we get new bounds for the energy E(G) = |λ_i| of G, where λ₁, λ₂, ..., λ_n are the eigenvalues of G (with multiplicity). Most of the results are proved for the more general situation of a Hermitian matrix whose characteristic polynomial has integral coefficients. This work was done during a visit of the second named author to UNAM.     

Constructions and bounds on linear error-block codes

We obtain new bounds on the parameters and we give new constructions of linear error-block codes. We obtain a Gilbert–Varshamov type construction. Using our bounds and constructions we obtain some infinite families of optimal linear error-block codes over . We also study the asymptotic of linear error-block codes. We define the real valued function α _q,m,a (δ), which is an analog of the important real valued function α _q (δ) in the asymptotic theory of classical linear error-correcting codes. We obtain both Gilbert–Varshamov and algebraic geometry type lower bounds on α _q,m,a (δ). We compare these lower bounds in graphs.      

Parity patterns associated with lifts of Hecke groups

Let q be an odd prime, m a positive integer, and let Γ _m (q) be the group generated by two elements x and y subject to the relations x ^2m =y ^qm =1 and x ²=y ^q ; that is, Γ _m (q) is the free product of two cyclic groups of orders 2m respectively qm, amalgamated along their subgroups of order m. Our main result determines the parity behaviour of the generalized subgroup numbers of Γ _m (q) which were defined in Müller (Adv. Math. 153:118–154, 2000), and which count all the homomorphisms of index n subgroups of Γ _m (q) into a given finite group H, in the case when gcd (m,| H |)=1. This computation depends upon the solution of three counting problems in the Hecke group ℋ(q)=C ₂*C _q : (i) determination of the parity of the subgroup numbers of ℋ(q); (ii) determination of the parity of the number of index n subgroups of ℋ(q) which are isomorphic to a free product of copies of C ₂ and of C _∞; (iii) determination of the parity of the number of index n subgroups in ℋ(q) which are isomorphic to a free product of copies of C _q . The first problem has already been solved in Müller (Groups: Topological, Combinatorial and Arithmetic Aspects, LMS Lecture Notes Series, vol. 311, pp. 327–374, Cambridge University Press, Cambridge, 2004). The bulk of our paper deals with the solution of Problems (ii) and (iii). Research of C. Krattenthaler partially supported by the Austrian Science Foundation FWF, grant S9607-N13, in the framework of the National Research Network “Analytic Combinatorics and Probabilistic Number Theory”.     

The existence of even cycles with specific lengths in Wenger’s graph

Wenger＇s graph Hm（q） is a q-regular bipartite graph of order 2qm constructed by using the mdimensional vector space Fq^m over the finite field Fq. The existence of the cycles of certain even length plays an important role in the study of the accurate order of the Turan number ex（n; C2m） in extremal graph theory. In this paper, we use the algebraic methods of linear system of equations over the finite field and the “critical zero-sum sequences” to show that： if m ≥ 3, then for any integer l with l ≠ 5, 4 ≤ l ≤ 2ch（Fq） （where ch（Fq） is the character of the finite field Fq） and any vertex v in the Wenger＇s graph Hm（q）, there is a cycle of length 21 in Hm（q） passing through the vertex v.